
Proof: We follow similar arguments as in Refs. [1]}, [2]}. Consider some LOCC operation in terms of its Kraus operators \(\lbrace  A_k \rbrace _{k}\) , with \(\sum _k A_k^\dagger A_k \le \mathbb {1}\) . This transforms the state \(\rho \)  into \(\sum _k p_k \rho _k^\prime \) , with \(p_k:=\text{Tr}[A_k \rho A_k^\dagger ]\)  and \(\rho _k^\prime := A_k \rho A^\dagger _k /p_k\) . We now compute
\(&\sum _k p_k\, \mathcal {E}_{\mathcal {M}}(\rho _k^\prime ) = \sum _k p_k \max \lbrace  0 , -\text{Tr}[W \rho _k^\prime ] \rbrace  \\& \ \ \ = - \sum _i \text{Tr}[W A_i\, \rho A_i^\dagger ]\,=\, - \sum _i \text{Tr}[A_i^\dagger W A_i\, \rho ] \\& \ \ \ = - \text{Tr}[W^\prime \rho ]\,\le \, \mathcal {E}_{\mathcal {M}}(\rho ), \) 
